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<h1 class="ltx_title ltx_title_section">PSF model: Rotated elliptical Gaussian function (3D using astigmatism)</h1>

<div id="Sx1.p1" class="ltx_para">
<p class="ltx_p">3D SMLM imaging can be performed by introducing a weak cylindrical
lens into the imaging path to create slight astigmatism in the image
<cite class="ltx_cite">[<a href="#bib.bib14" title="Three-dimensional super-resolution imaging by stochastic optical reconstruction microscopy" class="ltx_ref">2</a>]</cite>. This results in images of molecules with different
ellipticity depending on their axial position. When a molecule is
in focus, its image appears round. If the molecule is slightly above
or below the focal plane, its image appears ellipsoidal. Calibration
of the imaging system is needed to determine the orientation of the
imaged ellipsoid (the camera chip might not be aligned with cylindrical
lens) and the relationships between the axial position and ellipticity
of the imaged molecules.</p>
</div>
<div id="Sx1.p2" class="ltx_para">
<p class="ltx_p">A common PSF model for astigmatic 3D imaging is a rotated, elliptical
Gaussian function given by the formula</p>
<table id="Sx1.E1" class="ltx_equation">

<tr class="ltx_equation ltx_align_baseline">
<td class="ltx_eqn_pad"></td>
<td class="ltx_align_center"><img id="Sx1.E1.m1" class="ltx_Math" style="vertical-align:-20px" src="mi/mi50.png" width="454" height="48" alt="\mathrm{PSF}_{\mathrm{EG}}\left(x,y\mid\boldsymbol{\theta},\phi\right)=\frac{%
\theta_{N}}{2\pi\theta_{\sigma_{1}}\theta_{\sigma_{2}}}\exp\left(-\frac{x^{%
\prime}{}^{2}}{2\theta_{\sigma_{1}}^{2}}-\frac{y^{\prime 2}}{2\theta_{\sigma_{%
2}}^{2}}\right)+\theta_{b}\,,"></td>
<td class="ltx_eqn_pad"></td>
<td rowspan="1" class="ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation">(1)</span></td>
</tr>
</table>
<p class="ltx_p">where <img id="Sx1.p2.m1" class="ltx_Math" style="vertical-align:-6px" src="mi/mi61.png" width="146" height="21" alt="\mathrm{PSF}_{\mathrm{EG}}\left(x,y\mid\boldsymbol{\theta},\phi\right)">
gives the expected photon count at the integer pixel position <img id="Sx1.p2.m2" class="ltx_Math" style="vertical-align:-6px" src="mi/mi60.png" width="45" height="21" alt="\left(x,y\right)">
for the parameters <img id="Sx1.p2.m3" class="ltx_Math" style="vertical-align:-7px" src="mi/mi55.png" width="206" height="22" alt="\boldsymbol{\theta}=\left[\theta_{x},\theta_{y},\theta_{\sigma_{1}},\theta_{%
\sigma_{2}},\theta_{N},\theta_{b}\right]">,
and</p>
<table id="Sx1.EGx1" class="ltx_equationgroup ltx_eqn_eqnarray">

<tr id="Sx1.Ex1" class="ltx_equation ltx_align_baseline">
<td class="ltx_eqn_pad"></td>
<td class="ltx_td ltx_align_right"><img id="Sx1.Ex1.m1" class="ltx_Math" style="vertical-align:-2px" src="mi/mi52.png" width="19" height="18" alt="\displaystyle x^{\prime}"></td>
<td class="ltx_td ltx_align_center"><img id="Sx1.Ex1.m2" class="ltx_Math" style="vertical-align:-2px" src="mi/mi1.png" width="18" height="11" alt="\displaystyle="></td>
<td class="ltx_td ltx_align_left"><img id="Sx1.Ex1.m3" class="ltx_Math" style="vertical-align:-7px" src="mi/mi48.png" width="235" height="22" alt="\displaystyle\left(x-\theta_{x}\right)\cos\phi-\left(y-\theta_{y}\right)\sin%
\phi\,,"></td>
<td class="ltx_eqn_pad"></td>
</tr>
<tr id="Sx1.Ex2" class="ltx_equation ltx_align_baseline">
<td class="ltx_eqn_pad"></td>
<td class="ltx_td ltx_align_right"><img id="Sx1.Ex2.m1" class="ltx_Math" style="vertical-align:-5px" src="mi/mi54.png" width="18" height="21" alt="\displaystyle y^{\prime}"></td>
<td class="ltx_td ltx_align_center"><img id="Sx1.Ex2.m2" class="ltx_Math" style="vertical-align:-2px" src="mi/mi1.png" width="18" height="11" alt="\displaystyle="></td>
<td class="ltx_td ltx_align_left"><img id="Sx1.Ex2.m3" class="ltx_Math" style="vertical-align:-7px" src="mi/mi49.png" width="235" height="22" alt="\displaystyle\left(x-\theta_{x}\right)\sin\phi+\left(y-\theta_{y}\right)\cos%
\phi\,."></td>
<td class="ltx_eqn_pad"></td>
</tr>
</table>
<p class="ltx_p">The entries of the vector <img id="Sx1.p2.m4" class="ltx_Math" style="vertical-align:-2px" src="mi/mi56.png" width="15" height="16" alt="\boldsymbol{\theta}"> are as follows: <img id="Sx1.p2.m5" class="ltx_Math" style="vertical-align:-5px" src="mi/mi67.png" width="21" height="19" alt="\theta_{x}">
and <img id="Sx1.p2.m6" class="ltx_Math" style="vertical-align:-7px" src="mi/mi68.png" width="21" height="21" alt="\theta_{y}"> are the sub-pixel molecular coordinates, <img id="Sx1.p2.m7" class="ltx_Math" style="vertical-align:-6px" src="mi/mi64.png" width="28" height="20" alt="\theta_{\sigma_{1}}">
and <img id="Sx1.p2.m8" class="ltx_Math" style="vertical-align:-6px" src="mi/mi65.png" width="28" height="20" alt="\theta_{\sigma_{2}}"> are the imaged widths of the molecule along
two perpendicular axes rotated by the angle <img id="Sx1.p2.m9" class="ltx_Math" style="vertical-align:-5px" src="mi/mi12.png" width="15" height="19" alt="\phi"> with respect to
<img id="Sx1.p2.m10" class="ltx_Math" style="vertical-align:-5px" src="mi/mi69.png" width="23" height="15" alt="xy"> coordinates, <img id="Sx1.p2.m11" class="ltx_Math" style="vertical-align:-5px" src="mi/mi63.png" width="26" height="19" alt="\theta_{N}"> corresponds to the total number of
photons emitted by the molecule, and <img id="Sx1.p2.m12" class="ltx_Math" style="vertical-align:-5px" src="mi/mi66.png" width="20" height="19" alt="\theta_{b}"> is the background
signal level.
</p>
</div>
<div id="Sx1.SSx1" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">Estimating axial position</h2>

<div id="Sx1.SSx1.p1" class="ltx_para">
<p class="ltx_p">The estimate of the axial position <img id="Sx1.SSx1.p1.m1" class="ltx_Math" style="vertical-align:-2px" src="mi/mi59.png" width="13" height="16" alt="\hat{z}"> of a molecule is determined
by minimizing the distance between the fitted values <img id="Sx1.SSx1.p1.m2" class="ltx_Math" style="vertical-align:-6px" src="mi/mi58.png" width="60" height="25" alt="\hat{\theta}_{\sigma_{1}},\hat{\theta}_{\sigma_{2}}">
and the calibration curves <img id="Sx1.SSx1.p1.m3" class="ltx_Math" style="vertical-align:-6px" src="mi/mi57.png" width="102" height="21" alt="\hat{\sigma}_{1}\left(z\right),\hat{\sigma}_{2}\left(z\right)">,
obtained during the <a href="CalibrationEstimatorUI.html" title="" class="ltx_ref">calibration</a>
process, thus by</p>
<table id="Sx1.E2" class="ltx_equation">

<tr class="ltx_equation ltx_align_baseline">
<td class="ltx_eqn_pad"></td>
<td class="ltx_align_center"><img id="Sx1.E2.m1" class="ltx_Math" style="vertical-align:-19px" src="mi/mi47.png" width="433" height="46" alt="\hat{z}=\argmin_{z}\left(\left(\hat{\theta}_{\sigma_{1}}^{1/2}-\hat{\sigma}_{1%
}^{1/2}\left(z\right)\right)^{2}+\left(\hat{\theta}_{\sigma_{2}}^{1/2}-\hat{%
\sigma}_{2}^{1/2}\left(z\right)\right)^{2}\right)\,."></td>
<td class="ltx_eqn_pad"></td>
<td rowspan="1" class="ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation">(2)</span></td>
</tr>
</table>
<p class="ltx_p">Using the square root of the widths <img id="Sx1.SSx1.p1.m4" class="ltx_Math" style="vertical-align:-2px" src="mi/mi62.png" width="15" height="12" alt="\sigma"> slightly improves the
localization accuracy <cite class="ltx_cite">[<a href="#bib.bib14" title="Three-dimensional super-resolution imaging by stochastic optical reconstruction microscopy" class="ltx_ref">2</a>]</cite>. The minimization is performed
by the conjugate gradient method as implemented in the Apache Commons
Math library <cite class="ltx_cite">[<a href="#bib.bib6" title="The Apache Commons Mathematics Library; version 3.2" class="ltx_ref">1</a>]</cite> which is initialized from randomized
starting points to help avoid local minima.</p>
</div>
</div>
<div id="Sx1.SSx2" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">See also</h2>

<div id="Sx1.SSx2.p1" class="ltx_para">
<ul id="I1" class="ltx_itemize">
<li id="I1.i1" class="ltx_item" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">•</span> 
<div id="I1.i1.p1" class="ltx_para">
<p class="ltx_p"><a href="CalibrationEstimatorUI.html" title="" class="ltx_ref">Calibration of the imaging system for the astigmatism method</a></p>
</div>
</li>
<li id="I1.i2" class="ltx_item" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">•</span> 
<div id="I1.i2.p1" class="ltx_para">
<p class="ltx_p"><a href="PSF.html" title="" class="ltx_ref">Point-spread function (PSF)</a></p>
</div>
</li>
<li id="I1.i3" class="ltx_item" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">•</span> 
<div id="I1.i3.p1" class="ltx_para">
<p class="ltx_p"><a href="Fitting.html" title="" class="ltx_ref">Fitting point-spread function models</a></p>
</div>
</li>
<li id="I1.i4" class="ltx_item" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">•</span> 
<div id="I1.i4.p1" class="ltx_para">
<p class="ltx_p"><a href="FittingRegion.html" title="" class="ltx_ref">Definition of the fitting region</a></p>
</div>
</li>
<li id="I1.i5" class="ltx_item" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">•</span> 
<div id="I1.i5.p1" class="ltx_para">
<p class="ltx_p"><a href="LocalizationUncertainty.html" title="" class="ltx_ref">Localization uncertainty</a></p>
</div>
</li>
<li id="I1.i6" class="ltx_item" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">•</span> 
<div id="I1.i6.p1" class="ltx_para">
<p class="ltx_p"><a href="CrowdedField.html" title="" class="ltx_ref">Multiple-emitter fitting analysis</a></p>
</div>
</li>
</ul>
</div>
</div>
</div>
<div id="bib" class="ltx_bibliography">
<h1 class="ltx_title ltx_title_bibliography">References</h1>

<ul id="L1" class="ltx_biblist">
<li id="bib.bib6" class="ltx_bibitem ltx_bib_misc">
<span class="ltx_bibtag ltx_bib_key ltx_role_refnum">[1]</span>
<span class="ltx_bibblock"><span class="ltx_text ltx_bib_author">Commons-Math</span><span class="ltx_text ltx_bib_year">(2013-04)</span>
</span>
<span class="ltx_bibblock"><span class="ltx_text ltx_bib_title">The Apache Commons Mathematics Library; version 3.2</span>,
</span>
<span class="ltx_bibblock">External Links: <span class="ltx_text ltx_bib_links"><a href="http://commons.apache.org/proper/commons-math/" title="" class="ltx_ref ltx_bib_external">Link</a></span>.
</span>
<span class="ltx_bibblock ltx_bib_cited">Cited by: <a href="#Sx1.SSx1.p1" title="Estimating axial position ‣ PSF model: Rotated elliptical Gaussian function (3D using astigmatism)" class="ltx_ref"><span class="ltx_text ltx_ref_title">Estimating axial position</span></a>.
</span>
</li>
<li id="bib.bib14" class="ltx_bibitem ltx_bib_article">
<span class="ltx_bibtag ltx_bib_key ltx_role_refnum">[2]</span>
<span class="ltx_bibblock"><span class="ltx_text ltx_bib_author">B. Huang, W. Wang, M. Bates and X. Zhuang</span><span class="ltx_text ltx_bib_year">(2008)</span>
</span>
<span class="ltx_bibblock"><span class="ltx_text ltx_bib_title">Three-dimensional super-resolution imaging by stochastic optical reconstruction microscopy</span>,
</span>
<span class="ltx_bibblock"><span class="ltx_text ltx_bib_journal">Science</span> <span class="ltx_text ltx_bib_volume">319</span> (<span class="ltx_text ltx_bib_number">5864</span>), <span class="ltx_text ltx_bib_pages"> pp. 810–3</span>.
</span>
<span class="ltx_bibblock">External Links: <span class="ltx_text ltx_bib_links"><a href="http://dx.doi.org/10.1126/science.1153529" title="" class="ltx_ref doi ltx_bib_external">Document</a></span>.
</span>
<span class="ltx_bibblock ltx_bib_cited">Cited by: <a href="#Sx1.SSx1.p1" title="Estimating axial position ‣ PSF model: Rotated elliptical Gaussian function (3D using astigmatism)" class="ltx_ref"><span class="ltx_text ltx_ref_title">Estimating axial position</span></a>,
<a href="#Sx1.p1" title="PSF model: Rotated elliptical Gaussian function (3D using astigmatism)" class="ltx_ref"><span class="ltx_text ltx_ref_title">PSF model: Rotated elliptical Gaussian function (3D using astigmatism)</span></a>.
</span>
</li>
</ul>
</div>
</div>
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